// Requires CoherentSubgroups.mgm // In the introduction we make the following claim: every join-coherent // permutation group of degree at most 11 is either a cyclic group acting // regularly, a symmetric group, one of the groups described in Theorem 5, // or an imprimitive wreath product of join-coherent groups of smaller // degree as in Theorem 2(b). This includes the centralizers in Theorem 3. // This is easily seen directly for degree <= 4. For higher degrees it // is useful to use Magma to identify centralizers: these are always // join-coherent, and evidently they are imprimitive wreath products of // cyclic and symmetric groups of smaller degree. This leaves a small // number of other wreath products, and further groups to be identified. // Degree 5: // * Dihedral group order 10 (second case Theorem 5) // * Frobenius group order 20 (also second case Theorem 5) // * Cyclic group of order 5 acting regularly > Gs5, Cs5 := JoinCoherentSubgroups(5); > Gs5; [ Permutation group acting on a set of cardinality 5 Order = 10 = 2 * 5 (2, 5)(3, 4) (1, 4, 5, 2, 3), Permutation group acting on a set of cardinality 5 Order = 20 = 2^2 * 5 (1, 4, 3, 5) (1, 3)(4, 5) (1, 4, 2, 5, 3) ] > Cs5; [ [* Permutation group acting on a set of cardinality 5 Order = 5 (1, 4, 3, 2, 5), (1, 4, 3, 2, 5) *] ] // Degree 6: // * Wreath products C_2 wr C_3 and S_3 wr S_2 // * Centralizers of elements of cycle types (6), (3^2), (2^3) > Gs6, Cs6 := JoinCoherentSubgroups(6); > Gs6; [ Permutation group acting on a set of cardinality 6 Order = 24 = 2^3 * 3 (1, 6, 4)(2, 5, 3) (1, 3)(2, 6) (1, 3)(4, 5) (1, 3)(2, 6)(4, 5), Permutation group acting on a set of cardinality 6 Order = 72 = 2^3 * 3^2 (1, 4)(2, 6)(3, 5) (5, 6) (1, 6)(2, 4) (1, 6, 5)(2, 4, 3) (1, 6, 5)(2, 3, 4) ] > Cs6; [ [* Permutation group acting on a set of cardinality 6 Order = 6 = 2 * 3 (1, 5, 4)(2, 6, 3) (1, 2)(3, 4)(5, 6), (1, 6, 4, 2, 5, 3) *], [* Permutation group acting on a set of cardinality 6 Order = 18 = 2 * 3^2 (1, 3)(2, 5)(4, 6) (1, 5, 4)(2, 3, 6) (1, 4, 5)(2, 3, 6), (1, 4, 5)(2, 3, 6) *], [* Permutation group acting on a set of cardinality 6 Order = 48 = 2^4 * 3 (1, 5)(2 , 3)(4, 6) (1, 3, 6)(2, 5, 4) (3, 5)(4, 6) (1, 2)(3, 5) (1, 2)(3, 5)(4, 6), (1, 2)(3, 5)(4, 6) *] ] // Degree 7: // * All subgroups of Frobenius group of order 42 (second case Theorem 5) > Gs7, Cs7 := JoinCoherentSubgroups(7); > Gs7; [ Permutation group acting on a set of cardinality 7 Order = 14 = 2 * 7 (2, 4)(3, 6)(5, 7) (1, 3, 5, 4, 2, 7, 6), Permutation group acting on a set of cardinality 7 Order = 21 = 3 * 7 (1, 2, 4)(3, 7, 5) (1, 2, 7, 4, 3, 5, 6), Permutation group acting on a set of cardinality 7 Order = 42 = 2 * 3 * 7 (2, 3, 5)(4, 6, 7) (2, 4)(3, 6)(5, 7) (1, 3, 5, 4, 2, 7, 6) ] > Cs7; [ [* Permutation group acting on a set of cardinality 7 Order = 7 (1, 2, 7, 3, 6, 4, 5), (1, 2, 7, 3, 6, 4, 5) *] ] // Degree 8: // * Gamma(2^3) from the first case in Theorem 5 (see Proposition 10.1) // * Wreath products C_2 wr C_4, C_2 wr C_2 wr C_2 and S_4 wr S_2 // * Centralizers of elements of cycle type (8), (4^2), (2^4) > Gs8, Cs8 := JoinCoherentSubgroups(8); > Gs8; [ Permutation group acting on a set of cardinality 8 Order = 16 = 2^4 (1, 2, 3, 8, 5, 4, 6, 7) (1, 6, 5, 3)(2, 8, 4, 7) (2, 4)(7, 8) (1, 5)(3, 6), Permutation group acting on a set of cardinality 8 Order = 64 = 2^6 (7, 8) (1, 2, 3, 8, 5, 4, 6, 7) (3, 6)(7, 8) (2, 4)(7, 8) (1, 6)(2, 8)(3, 5)(4, 7) (1, 5)(3, 6), Permutation group acting on a set of cardinality 8 Order = 128 = 2^7 (7, 8) (1, 2, 5, 4)(3, 8)(6, 7) (3, 6)(7, 8) (2, 4)(7, 8) (2, 8)(4, 7) (1, 5)(3, 6) (1, 6)(3, 5), Permutation group acting on a set of cardinality 8 Order = 1152 = 2^7 * 3^2 (7, 8) (1, 2, 5, 4)(3, 8)(6, 7) (1, 3)(7, 8) (4, 8, 7) (1, 6, 3) (2, 4)(7, 8) (2, 8)(4, 7) (1, 5)(3, 6) (1, 6)(3, 5) ] // Checking assertions above: > S8 := SymmetricGroup(8); > Gs8[2] eq sub; > Gs8[3] eq sub; > Gs8[4] eq sub; > Cs8; [ [* Permutation group acting on a set of cardinality 8 Order = 8 = 2^3 (1, 2, 3, 6, 5, 7, 4, 8) (1, 4, 5, 3)(2, 8, 7, 6) (1, 5)(2, 7)(3, 4)(6, 8), (1, 2, 3, 6, 5, 7, 4, 8) *], [* Permutation group acting on a set of cardinality 8 Order = 32 = 2^5 (2, 6, 7, 8) (1, 2, 3, 6, 5, 7, 4, 8) (1, 4, 5, 3)(2, 6, 7, 8) (2, 7)(6, 8) (1, 5)(3, 4), (1, 3, 5, 4)(2, 6, 7, 8) *], [* Permutation group acting on a set of cardinality 8 Order = 384 = 2^7 * 3 (3, 8)(4, 6) (2, 4, 6)(3, 8, 7) (1, 3)(2, 6)(4, 5)(7, 8) (6, 8) (1, 7)(2, 5)(3, 8)(4, 6) (2, 7)(6, 8) (3, 4)(6, 8) (1, 5)(2, 7)(3, 4)(6, 8), (1, 5)(2, 7)(3, 4)(6, 8) *] ] // Degree 9: // * Gamma(3^2) from first case in Theorem 5 (see Proposition 10.1) // * Wreath products C_3 wr C_3, S_3 wr C_3, S_3 wr S_3 // * Centralizers of elements of cycle type (9) and (3^3) > Gs9, Cs9 := JoinCoherentSubgroups(9); > Gs9; [ Permutation group acting on a set of cardinality 9 Order = 27 = 3^3 (1, 7, 4, 3, 8, 5, 2, 9, 6) (1, 3, 2)(7, 9, 8) (4, 5, 6)(7, 9, 8), Permutation group acting on a set of cardinality 9 Order = 81 = 3^4 (1, 8, 5)(2, 7, 4)(3, 9, 6) (1, 3, 2)(7, 8, 9) (4, 5, 6)(7, 8, 9) (7, 8, 9), Permutation group acting on a set of cardinality 9 Order = 648 = 2^3 * 3^4 (8, 9) (1, 8, 5)(2, 7, 4)(3, 9, 6) (5, 6)(8, 9) (2, 3)(4, 5, 6)(8, 9) (1, 3, 2)(7, 8, 9) (4, 5, 6)(7, 8, 9) (7, 8, 9), Permutation group acting on a set of cardinality 9 Order = 1296 = 2^4 * 3^4 (1, 2, 3) (4, 5, 6) (1, 5, 2, 4)(3, 6) (1, 4, 7)(2, 5, 8)(3, 6, 9) (1, 7, 2, 8)(3, 9)(4, 5) ] // Checking assertions above: > S9 := SymmetricGroup(9); > Gs9[2] eq sub; > Gs9[3] eq sub; > Gs9[4] eq sub; > Cs9; [ [* Permutation group acting on a set of cardinality 9 Order = 9 = 3^2 (1, 7, 5, 3, 8, 6, 2, 9, 4) (1, 3, 2)(4, 5, 6)(7, 8, 9), (1, 7, 5, 3, 8, 6, 2, 9, 4) *], [* Permutation group acting on a set of cardinality 9 Order = 162 = 2 * 3^4 (4, 7, 6, 8, 5, 9) (1, 7, 4, 2, 8, 6, 3, 9, 5) (4, 5, 6)(7, 8, 9) (1, 3, 2)(7, 8, 9) (7, 8, 9), (1, 3, 2)(4, 5, 6)(7, 9, 8) *] ] // Degree 10: // * Wreath products D_5 wr C_2, C_2 wr C_5, C_2 wr D_5, F_20 wr C_2, C_2 wr F_20 // * Centralizers of elements of cycle type (10), (5^2) and (2^5) // where D_5 denotes the dihedral group of order 10 and F_20 the Frobenius // group of order 20 (see degree 5 classification) > Gs10, Cs10 := JoinCoherentSubgroups(10); > Gs10; [ Permutation group acting on a set of cardinality 10 Order = 200 = 2^3 * 5^2 (6, 10)(7, 8) (1, 6)(2, 9)(3, 7)(4, 8)(5, 10) (1, 5)(3, 4)(6, 10)(7, 8) (1, 5, 4, 2, 3) (6, 8, 7, 10, 9), Permutation group acting on a set of cardinality 10 Order = 160 = 2^5 * 5 (1, 9, 3, 6, 8)(2, 10, 4, 5, 7) (1, 2) (3, 4) (5, 6) (7, 8) (9, 10), Permutation group acting on a set of cardinality 10 Order = 320 = 2^6 * 5 (1, 8)(2, 7)(5, 10)(6, 9) (1, 9, 3, 6, 8)(2, 10, 4, 5, 7) (1, 2) (3, 4) (5, 6) (7, 8) (9, 10), Permutation group acting on a set of cardinality 10 Order = 800 = 2^5 * 5^2 (7, 8, 9, 10) (1, 7, 5, 10, 3, 9, 4, 8)(2, 6) (7, 9)(8, 10) (1, 5, 3, 4)(7, 8, 9, 10) (1, 3)(4, 5)(7, 9)(8, 10) (6, 7, 8, 10, 9) (1, 3, 4, 2, 5), Permutation group acting on a set of cardinality 10 Order = 640 = 2^7 * 5 (1, 2)(3, 6, 9, 8, 4, 5, 10, 7) (1, 2)(3, 9)(4, 10)(5, 7, 6, 8) (1, 7, 9, 3, 6)(2, 8, 10, 4, 5) (1, 2) (3, 4) (5, 6) (7, 8) (9, 10) ] // Checking assertions above: > S10 := SymmetricGroup(10); > Gs10[1] eq sub; > Gs10[2] eq sub; > Gs10[3] eq sub; > Gs10[4] eq sub; > Gs10[5] eq sub; > Cs10; [ [* Permutation group acting on a set of cardinality 10 Order = 10 = 2 * 5 (1, 9, 3, 6, 8)(2, 10, 4, 5, 7) (1, 2)(3, 4)(5, 6)(7, 8)(9, 10), (1, 10, 3, 5, 8, 2, 9, 4, 6, 7) *], [* Permutation group acting on a set of cardinality 10 Order = 50 = 2 * 5^2 (1, 6)(2, 7)(3, 8)(4, 10)(5, 9) (6, 7, 10, 8, 9) (1, 4, 5, 2, 3), (1, 2, 4, 3, 5)(6, 7, 10, 8, 9) *], [* Permutation group acting on a set of cardinality 10 Order = 3840 = 2^8 * 3 * 5 (1, 3, 5, 7, 9)(2, 4, 6, 8, 10) (1, 3)(2, 4) (1, 2), (1, 2)(3, 4)(5, 6)(7, 8)(9, 10) *] ] // Degree 11: // * All subgroups of Frobenius group of order 110 (second case Theorem 5) // * Cyclic group of order 11 acting regularly > Gs11, Cs11 := JoinCoherentSubgroups(11); > Gs11; [ Permutation group acting on a set of cardinality 11 Order = 22 = 2 * 11 (2, 11)(3, 10)(4, 9)(5, 8)(6, 7) (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), Permutation group acting on a set of cardinality 11 Order = 55 = 5 * 11 (2, 5, 6, 10, 4)(3, 9, 11, 8, 7) (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), Permutation group acting on a set of cardinality 11 Order = 110 = 2 * 5 * 11 (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) (1, 2, 4, 8, 5, 10, 9, 7, 3, 6) ] > Cs11; [ [* Permutation group acting on a set of cardinality 11 Order = 11 (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) *] ] // Degree 12 example from introduction S12 := SymmetricGroup(12); G := sub; testG1 := TestSubgroup(Join,G); testG2 := G subset sub;